SpinyHand: Contact Load Sharing for aHuman-Scale Climbing Robot

Shiquan Wang∗ †

Hao Jiang∗Tae Myung Huh

Danning SunWilson RuotoloMatthew Miller

Will R. T. RoderickGraduate Research Assistant

Dept. of Mechanical EngineeringStanford University

424 Panama Mall, Bldg 560Stanford, California, 94305 USA

Hannah S. StuartAssistant Professor

Dept. of Mechanical EngineeringUniversity of California2521 Hearst Avenue

Berkeley, California, 94709 USAhstuart@berkeley.edu

Mark R. CutkoskyProfessor

Dept. of Mechanical EngineeringStanford University

416 Escondido Mall, Bldg 550Stanford, California, 94305 USA

cutkosky@stanford.edu

We present a hand specialized for climbing unstructuredrocky surfaces. Articulated fingers achieve grasps commonlyused by human climbers. The gripping surfaces are equippedwith dense arrays of spines that engage with asperities onhard rough materials. A load-sharing transmission systemdivides the shear contact force among spine tiles on eachphalanx to prevent premature spine slippage or grasp failure.Taking advantage of the hand’s kinematic and load-sharingproperties, the wrench space of achievable forces and mo-ments can be computed rapidly. Bench-top tests show agree-ment with the model, with average wrench space errors of10-15%, despite the stochastic nature of spine/surface inter-action. The model provides design guidelines and controlstrategy insights for the SpinyHand and can inform futurework.

1 IntroductionThe ability of robots to climb walls, cliffs and other

steep surfaces would be beneficial for various applicationsincluding planetary exploration and search and rescue. In

∗At the time of publication, this author is affiliated with Flexiv Robotics,Ltd., 4500 Great America Pkwy, Santa Clara, Califronia 95054.

†Address all correspondences to this author: shiquan.wang@flexiv.com.

recent years, a number of climbing robots have used clawsor arrays of miniature spines to climb steep and even verticalsurfaces like concrete and stucco walls [1–12]. Most of theseplatforms are limited to relatively uniform surfaces. A largerobot, with long legs, can traverse complex surfaces by step-ping over obstacles and bridging gaps. However, it becomesincreasingly challenging to obtain sufficient traction with in-creasing robot size, denoted by L, because mass grows as L3

while foot contact area grows as L2. Failure can be catas-trophic for a large climbing robot when it falls from heightsof just a meter.

SpinyHand is part of a collaboration among Duke Uni-versity, Stanford University, and the University of CaliforniaSanta Barbara to enable the 100 kg RoboSimian [13] to climbsteep rocky surfaces. To support large loads on surfaces thatare not relatively flat, the design presented here and shown inFig. 1 combines spine arrays with an anthropomorphic, mul-tifinger hand that is articulated to mimic rock climber strate-gies. The spines do not penetrate the surface but interact withsurface features to provide large contact forces.

In the following sections, we briefly introduce the spinetechnology utilized in this work and review current multi-finger grasp modeling strategies. Section 2 introduces theSpinyHand design, including the classification of graspingstrategies inspired by human rock-climbers. Passive load

Wang JMR-18-1464 1

Fig. 1: SpinyHand grasping a pumice rock. The contact areas are coveredin microspines to support large tangential forces.

sharing among the contacts allows the hand to sustain highexternal forces and moments on convex surfaces. Design ofthe fingers and palm is described in Sections 3 and 4 respec-tively, with special attention given to spine engagement. Sec-tion 5 extends the analysis for predicting overall force andmoment capabilities as a function of finger parameters andthe curvature of the grasped surface. Initial model valida-tion is conducted with bench-top experiments in Section 6.Trials focus on low curvature surfaces, for which the spinesprovide the greatest advantage in comparison to conventionalgrippers or hands. Section 7 discusses additional design andcontrol considerations, using simulated examples. Section 8gives recommendations for future work.

1.1 Spines for ClimbingSpines are an evolving technology that enable robots to

attach to hard, rough surfaces. Inspired by the tarsal spinesof insects [14], they can catch on small asperities (bumpsand pits) on surfaces. On aggregate, over many spines, thisengagement can sustain large contact forces. The climbingrobots cited in the last section often use elastic mechanismsthat allow each spine to settle on an asperity and distributethe overall load force among groups of spines [1,15]. How-ever, when scaling to large robots this approach requires im-practically large contact areas, which become awkward tocontrol and can prevent attachment to contoured surfaces.

A higher density of spine contact is therefore required.As modeled in [16], linearly-constrained spines consist ofarrays of small, sharp spikes that slide freely normal to thesubstrate, allowing them to conform to local contours. Densepacking allows for high contact stresses, although the varia-tion in loading between spines may be higher than with pre-vious designs. Unlike typical frictional contact, the spinearrays provide high shear forces on rough surfaces with littleor no normal force applied. A patch of spine-equipped tileswith half the surface area of the SpinyHand presented in thispaper could hold up to 70kg on a flat concrete wall [17].

1.2 Multifinger GraspingMultifinger grasping has been studied extensively; for

recent reviews of grasp analysis and hand design, see [18]and [19], respectively. Much of the grasping literature fo-cuses on identifying contact locations and grasp forces toachieve a stable or optimal grasp [20–22] but the mechanicsof spine contact fundamentally alter grasping performanceand analysis. While the hand must still conform to surfaces,grasp strength – the hand’s ability to resist pull-out forces– no longer depends on frictional contacts (Ft ≤ µFn). In-stead, the maximum tangential force at each spine-array isa function of contact conditions (e.g. roughness) and ma-terial strength; grasp strength is nearly unchanged over awide range of positive normal contact forces, as describedin [16,17,23]. It follows that this work focuses on distribut-ing tangential, not normal, forces in Section 3.1 and 4.1. Thistangential load sharing is accomplished with a single tendonloading all spine tiles on each finger. Although the objectiveis different, similar concepts have been presented for bal-ancing torques among the joints of an underactuated hand(e.g. [24,25]).

The basic mathematical framework in the literature fordescribing a grasp in terms of links, joints, contacts, and ten-don tensions remains valid and useful. The central challengeconsists of an optimization problem: one must simultane-ously minimize the energy of the hand system while alsosatisfying contact geometry and force constraints. Solversfor such problems typically require convex constraints, suchas Coulomb friction contact. This requirement is not satis-fied by spine tile contact, for which the loading limits form anon-convex boat-shaped hull [16]. One computational solu-tion uses a mixed-integer linear programming method, accel-erated by branch-and-bound and hierarchical convex decom-position methods [26]. Alternatively, in this paper, we takeadvantage of the load-sharing design of the fingers such thatthe tangential force at each contact is known immediatelyfrom the tendon tension. This simplifies the grasp analysis,as described in Section 5.4.

2 Overall DesignThe hands must robustly grip under heavy loading con-

ditions of up to 50 kg (50% of RoboSimian’s total mass). Inorder to do so, they should provide: (i) adaptability to a widevariety of terrain features and (ii) good contact quality on adiverse set of surfaces.

Human rock climbers adapt to a variety of terrains;sloper (open-hand), crimp and pinch gripping techniqueswork effectively on geological features such as roundedslopes, cracks, edges, and ridges [27–30]. As shown in Fig.2, the four-finger anthropomorphic SpinyHand can performall the aforementioned grasps. The under-actuated fingersare each curled by a single tendon. They switch from a par-allel (sloper and crimp) pose to an opposed (pinch) config-uration by rotating the two outer fingers 180◦ at the base.Articulated fingernails help to cling on shallow ledges.

Hands disproportionately affect the inertia of a robot’slimbs, so designers often reduce weight by using differen-

Wang JMR-18-1464 2

Fig. 2: (left) SpinyHand CAD rendering shows 22 spine-laden tiles in yellow distributed among 4 independent fingers and a palm. (right) Rock climbingtechniques (a-d): (a) sloper grasp on a gently curved surface; (b) open hand/half crimp grasp on an edge; (c) pinch grasp on a narrow feature; (d) posebetween a sloper grasp and pinch. (d) is not commonly used by human rock climber but can be helpful for SpinyHand to envelop large-curvatures.

Fig. 3: Spine tile design: (1) sliding block for prismatic motion of the fingerspine tile along x axis; (2) fixed pulley (horizontal metal rod); (3) groove fortendon routing; (4) sliding channel for palm tile; (5) fixed pulley (verticalmetal rod). The red arrow indicates the sliding and loading direction. Thetile coordinates are consistent with the finger phalanx coordinates P.

tials or clutches and multiplexing mechanisms to drive mul-tiple fingers with fewer motors. However, when consideringthe benefits of brakes or clutches one must also consider howthe hand responds when a grasp failure occurs. Spinyhand’sfingers must react immediately and independently to asperitybreakage and slippage, allowing the posture to readjust with-out detaching and causing the robot to fall. In the case ofSpinyHand, a separate 20 W motor is dedicated to each fin-ger with a 236:1 gear ratio that provides a maximum tendontension of approximately 700 N, accounting for friction.

Although the transmission ratio is relatively high, thereis enough compliance in the tendon and tile system to reducethe chances of tendon breakage when slips or impacts occur.The fingers and palm are machined from 7075 Al alloy andare sufficiently robust to survive impacts and grasp failures.The entire hand has a mass of 2.6 kg and each finger canproduce up to 600 N at each fingertip. For implementationdetails, see Appendix 9.1.

When climbing on large rocky features, a hand is un-likely to achieve grasp form-closure; contact properties be-tween the hand and the surface ultimately limit grip secu-rity. Most existing anthropomorphic hands use patches offrictional material to improve the contact interface, and pushor squeeze on surfaces to achieve force-closure. This ap-proach is insufficient for a 100 kg robot “palming” verticalfeatures because lateral forces are very high, and squeezingharder pushes the robot away from the wall. The linearly-constrained spines on SpinyHand engage with many smallsurface asperities, and can provide large normal and shearattachment forces [16]. The hand is covered with twenty-

two spine tiles (pictured in Fig. 3), spread throughout thepalm and fingers. Tendon-based loading mechanisms dis-tribute contact forces, as described in the next section.

3 Spined Fingers3.1 Prismatic Phalanx

Four identical tendon-driven fingers with pin joints pro-vide high torsional rigidity, a slender form factor, and a wideextended pose in order to grip flat surfaces. The inner surfaceof each phalanx is covered with spines, which most effec-tively provide attachment when also loaded in shear [16]. Assuch, the actuated contact forces should have a lateral com-ponent directed along the finger, similar to grippers that usedirectional adhesion [31,32].

As shown in Fig. 4, each phalanx consists of a frameand a sliding spine tile. Linear travel along each phalanx al-lows the tiles to catch asperities. A pair of extension springspreload the tile proximally when the fingers are open, butbias the tile distally when the tile is actuated against a sur-face. The tile’s pulley axis (a fixed metal rod on the spinetile) is parallel to the finger joint axis, and a groove on thetile surface provides a path for the tendon. A single tendonsequentially routes around both the joint and tile pulleys, ac-tuating spine and finger motions simultaneously. The shearcomponent of each contact thus grows proportionally withthe finger joint torque and tendon tension. Contact loadingconditions, such as the ratio between shear and normal com-ponents, are determined by pulley sizes. This mechanisminfluences grasp modeling in Section 5.

3.2 Fingernail and finger closing sequencesAn articulated fingernail, built into the distal end of each

digit, is designed for clinging on very shallow edges or fingerholes where the main phalanx cannot fit. Structured aroundanother pin joint, it is preloaded by a pair of soft springs.To perform clinging, fingernail actuation occurs while thefingers are straight. With high-static-load joint bushingsand sufficient structural strength of spine tiles and phalanxframes (verified with FEA), the entire finger can sustain morethan 500 N shear load when clinging to terrain features withthe fingernail. To prevent this ultrashort phalanx from inter-fering with the main finger engagement, pushing the distalphalanx away from the surface, the fingernail rotates a full180◦ to retract before the tendon tension is large enough to

Wang JMR-18-1464 3

Fig. 4: Diagram and pictures of the finger: (1) fingernail; (2) spine tile; (3)joint pulley; (4) phalanx extension spring; (5) phalanx frame; (6) phalanxpulley; (7) joint extension spring. Ft is the tendon force. The tendon (red)routes around each joint pulley and phalanx pulley, so that Ft simultaneouslycurls the finger and slides the spine tile along the phalanx.

actuate the rest of finger joints.After the fingernail retracts and hits its hard stop, the

rest of finger starts to curl starting from the proximal to distalphalanges. This approach, largely implemented with springelements in the fingers, prevents grasp ejection due to prema-ture distal curling [33,34]. Only after all fingers have settledon the surface should the prismatic phalanges start to slideand load the surface, otherwise the spine tiles can no longerbe actuated to generate shear contact forces. The fingernailjoint spring is the softest (0.025 Nm/rad). Joint springs areslightly stiffer (0.05 Nm/rad). They must be strong enoughto prevent finger sagging due to gravity and inertia, but al-low full finger closing before the prismatic phalanges slide.Spine-tile springs are the strongest (11.5 N/mm), yet mustnot inhibit prismatic motion given actuator capabilities andmotion requirements for reliable surface engagement. Springpreloads can also be used to tune the motion sequences with-out diminishing strength, as noted in [35].

4 Spined PalmBased on the spine array in [17], the SpinyHand palm

is large, flat and covered with spine tiles (11× 6 cm withover 300 spines). Shafts support the eight tiles while al-lowing them to slide in the palm. Springs hold the tiles inplace until spines catch on usable terrain features. A sin-gle tendon routes around all pulleys on the spine tiles andpalm in order to distribute the load to all spine tiles relativelyevenly. However, because the pulleys are polished stainlesssteel rods, there is some friction, which produces some varia-tion in loading. As seen in the next section, a small amount offriction is actually desirable to increase robustness becauseit allows stiffer and stronger tile-surface contacts to assumemore of the load.

4.1 Load-sharing with frictionLoad sharing creates the effects that spines are additive

and that tiles which happen to catch asperities on first contactdo not take the brunt of the load. The strategy of attaching

Fig. 5: Left: Palm experiment setup. Right: diagram of a 2-tile system.Parts with boundaries or shades in the same color are mounted together assingle moving parts. Two tiles in gray are preloaded by springs (stiffnesskp) mounted to a moving palm frame in red. The tiles have contact stiffnessk1 and k2 with respect to the ground in blue. A tendon routes over pulleyson the tile (gray) and palm frame (red). Local tendon tensions Ft1 and Ft2may be different due to friction on pulleys between adjacent tiles. x1, x2 andx are global displacements of the tiles and palm frame respectively.

multiple tiles to a single tendon would, in the case of zerofriction, produce exactly the same tangential force at eachtile. With friction, this is no longer the case.

For simplicity, two adjacent spine tiles are modeled, rep-resented in Fig. 5. Each tile (grey) is preloaded by springs(black, with stiffness kp) anchored to the palm (red) and en-gaged to the surface via a contact stiffness (black, with stiff-nesses k1, k2). The loading force on the palm, F , results intendon tensions (Ft0, Ft1, and Ft2) and displacements of palmand tiles (x, x1, and x2), respectively. The coefficient of fric-tion associated with each pair of tendon tensions is denotedby µ and assumed constant. Given all the spring stiffnesses,four equations describe the force balance for the two tiles andtwo pairs of tendon tensions. Palm and tile displacements arealso constrained by the total tendon length. A relation can befound between any two unknowns among x, x1, x2, Ft0, Ft1and Ft2). Note that the tendon force balancing equation isfriction-direction dependent. Details of the derivation can befound in Appendix 9.2.

Solving the equations for contact forces as a function ofoverall palm travel, x, gives:

F1 =

2(2+µ)k1k2+4kpk1(2−µ)k1+(2+µ)k2+4kp

x

F2 =2(2−µ)k1k2+4kpk2

(2−µ)k1+(2+µ)k2+4kpx

(1)

We assume that the tile with less surface engagement, andthus lower contact stiffness, will limit the pair’s overall per-formance. The tendon/pulley friction, µ, causes the load toshift to the stiffer contact, which has caught the surface moresecurely. Ideally, F1/k1 = F2/k2 in order to distribute theload according to each tile’s force limit. The desired coeffi-cient of friction becomes:

µ = 2k1− k2

k1 + k2(2)

For a pair of tiles with a 2:1 force limit ratio, moderate fric-

Wang JMR-18-1464 4

tions with µ no larger than 2/3 help distribute the load moreappropriately. We can estimate the overall 2-tile palm forcelimited F by the worst engaged tile’s displacement, indicatedhere as x2:

F =4k1k2 +2kpk1 +2kpk2

(2−µ)k1 +2kpx2. (3)

4.2 Experiments and ResultsTwo- and four-tile load sharing systems are used to val-

idate the effect of transmission friction. As shown in Fig.5 for the four-tile case, interchangeable compression springsmounted between a grounded acrylic board (blue) and thetiles (grey) simulate contact stiffnesses. An acrylic palmframe (red) holds the transmission pulleys and extensionsprings. All pulleys can be switched from fixed rods (µempirically determined to be 0.34 for Dyneema tendons) tobearing-supported pulleys (µ is negligible). The force limitis simulated with a hard stop on the softest contact spring.Combinations of three contact stiffness springs are used: 1.4,2.2 and 4.9N/mm. As shown in Fig. 6, experimental resultswith the two-tile system are consistent and match the modelpredictions within 5%. Introducing increased pulley friction,improves maximum load by at least 14%.

The model for systems with more tiles is complex dueto the undetermined friction direction at each pulley; moder-ate tendon friction improves load sharing only between eachadjacent pair of tiles. Nonetheless, friction is helpful, par-

Fig. 6: Predicted and experimental maximum load of a two-tile system.Horizontal axis shows the contact spring configurations (springs 1, 2 and 3have stiffness of 1.4, 2.2 and 4.9N/mm respectively). Five measurementsare conducted for each spring configuration. The percentage in red showshow much friction can improve the loading capacity.

Fig. 7: Empirical maximum load of a four-tile system with and withoutfriction. Horizontal axis shows the contact stiffness spring configurations,with the spring type sequence starting from the left.

ticularly when there are substantial discrepancies in contactquality across all tiles. For example, in experiments for a 4-tile system (Fig. 7), tendon friction improves loading limitsin all 8 configurations by at least 24%.

5 Modeling Multifinger Grasping with SpinesGrasp strength estimation can guide new hand designs,

selection of terrain handhold techniques, hand control, androbot motion planning. However, spine attachment producesnon-convex frictional contacts, a property that makes graspmodels difficult to solve [36]. This section presents an ef-ficient approach to 3D modeling grasping with SpinyHand.We start by computing contact forces for given actuationforces, ignoring internal friction in the hand. We then con-sider the effect of friction in the hand in Section 5.2.

5.1 Contact ForcesThe first step is solving contact forces on a single Spiny-

Hand finger given actuation (tendon) force, finger design,and grasp surface geometry. Assuming no friction in thehand and a single contact on each phalanx, a simplifieddiagram demonstrates how contacts are distributed duringgrasping (Fig. 8). Three coordinate systems are used in themodel: (1) palm coordinates (global, G); (2) finger coordi-nates with the z-axis the same as that of the palm coordinatesand with x-z on the finger plane (F); (3) phalanx coordinateslocated at each phalanx (P). The model in this section is inthe x-z finger plane.

Utilizing geometry constraints of the surface, joint po-sitions of an n-phalanx finger q and the corresponding con-tact locations c can be computed with a bottom-up approach,from the proximal to distal phalanx. Let q = [q1 q2 . . .qn]

T bethe joint positions and c = [c1 c2 . . .cn]

T be the contact posi-tions, where ci is the position along the x-axis of the phalanx

Fig. 8: Three coordinates are used in the hand model: 1) palm (global)coordinates G; 2) finger coordinates F ; 3) phalanx coordinates P. Bluedots indicates the contact locations. ci and qi are contact forces and jointangles. A third finger is behind the grasping surface and therefore displayedwith dotted lines. The top diagrams show: (a) crimp grasp with only distalphalanx contact; (b) pinch/sloper grasp with proximal contacts broken.

Wang JMR-18-1464 5

frame i.1

At equilibrium, torques due to contact forces, tendon ac-tuation and joint springs should balance:

J(q,c)T f c + ftr−Krq− pr = 0 (4)

where J is an expanded Jacobian matrix [J1 J2 . . .Jn], with Jibeing the Jacobian matrix with respect to contact on the ith

phalanx. f c is a column list of the contact forces f (i)c (columnvector) for phalanx i. ft is a scalar value of the tendon force.r and pr are, respectively, column vectors of pulley radiusand preload torque for each joint. The joint stiffness matrixKr is a diagonal matrix containing all of the joint spring stiff-nesses of the elements being considered.

The floating tiles on each phalanx introduce anothergroup of force balancing constraints:

Px f c + f p(c, ft)−Kpd− pp = 0 (5)

where Px is a projection matrix for contact force on each pha-lanx in the x-axis of its local coordinates. f p is a columnvector of the forces along the x-axis of the spine tile due totendon actuation. Note that this force may change as the pha-lanx slides and affects the tendon angles on the tile pulley. dis a column of prismatic phalanx travels from their initial po-sitions. Similar to the structure of Eqn. 4, Kp and pp denotetile spring stiffness and preload force on each phalanx. Theabove equations can be rewritten compactly as:

A f c = b (6)

where

A =

[JT

Px

]and b =

[Krq+ pr− ftr

Kpd + pp− f p(c, ft)

](7)

Matrix A is square and invertible, therefore contactforces f c and can be solved for efficiently. We note that theprismatic phalanx design enables this approach and that so-lutions do not depend on the structural stiffness of the hand,which is often difficult to characterize.

5.2 Finger FrictionThe above model neglects friction within the hand. In

SpinyHand, friction is primarily in the tile pulleys. As thefingers close, this friction reduces the tendon tension ft se-quentially along the finger, from proximal to distal. Themagnitude of the friction force is proportional to tendon ten-sion and can be included by modifying b:

b =

[Krq+ pr− ftMr

Kpd + pp−M f p(c, ft)

](8)

1The subscript denotes finger/joint number starting at the proximal endof the chain. The same convention applies to other variables. Every up-per case, bold lower case and regular lower case symbol represents matrix,vector and scalar respectively.

where M is a friction matrix with diagonal elements of[1,(1−µ), . . .(1−µ)n−1], and µ is the lumped coefficient offriction between adjacent phalanges.

Without friction, any change of load will cause the fingerto adjust its posture, moving the tendon. With friction, themotions are resisted so that Eqn. 5 is not satisfied. Before thetendon starts to slide friction should balance any change inthe tangential contact forces d fci along the local x axis. Thefriction matrix in Eqn. 8 should be modified: each diagonalelement is replaced by (1+ d fci/ fpi) and bounded by (1±µ) when the tendon starts to slide. Appendix 9.3 providesdetails on solving for d fci.

5.3 Finger and Grasp StiffnessThe wrench of a finger settled on a surface with re-

spect to the global coordinate frame w f is computed with thewrench matrix Wf ; 2D forces on the finger plane are trans-formed to a 6D wrench with respect to the palm:

w f =Wf f c (9)

Small displacements of the 6D finger position x f (fingerbase, origin of finger coordinates) affect w f . The graspwrench wg is the sum of all finger wrenches at the wrist:

wg =n

∑i

w f i (10)

where w f i is wrench of the ith finger in a n-finger hand.Grasp and finger stiffness matrices express the effects of

small motions at the wrist and fingers on forces at the wristand finger contacts, respectively [37]. The finger stiffnessmatrix, K f = ∂w f /∂x f , is a combination of two elements:motions in the plane of the finger and structural stiffness outof the plane. Finger stiffness components are numericallycomputed based on changes in f c and w f (using Eqn. 6and Eqn. 9) as finger position x f is infinitesimally shiftedalong different finger DOFs (x, z, and θy in finger coordi-nates). Tendon force ft may depend on x f when computingK f , depending upon different tendon actuation schemes (e.g.position control). Structural stiffness components along theother three DOFs in finger coordinates are simplified by as-suming linear beam elasticity. The grasp stiffness matrix Kgis defined similarly to K f , with grasp position xg defined asthe palm coordinates.

5.4 Grasp Wrench and Wrist Motion5.4.1 Forward solution

Given the configuration of a grasp xg (6D positionand orientation), it is straightforward to compute the graspwrench wg using Eqn. 6, 8, 9 and 10, given actuator forces.

5.4.2 Inverse solutionSolving in reverse from wg to xg is useful for grasp plan-

ning but more complicated; small changes in forces or mo-tions applied at the wrist can produce substantial changes in

Wang JMR-18-1464 6

the contact forces and compliant deflections at the contacts,The gradient of wg with respect to xg can be represented withnumerical estimates of K f i and J f i:

dwg =n

∑i=1

K f iJ f idxg = Kgdxg (11)

where J f i is the twist transformation matrix [22] from palmcoordinates to coordinates of the ith finger:

dx f i = J f idxg (12)

and dx f i and dxg are the twists of the finger and the hand, re-spectively. Finger positions and contact forces can be solvedwith a gradient descent search; increments of xg are com-puted iteratively by inverting the grasp stiffness matrix withnumerical estimation.

The gradient descent search applied in this backward ap-proach may not converge for certain hand configurations, be-cause matrix A in Eqn. 6 is ill-conditioned when solvingfor K f (cond(A) > 270 in all trials). A small change in thefinger’s pose (finger base position x f ) can result in a largechange in the contact forces f c, as noted in the experimentsin Section 6. Smaller search steps typically resolve this issue,but the backward approach is less efficient than the forwardapproach due to its search-based solver and step constraint.

5.5 Grasp failureWe define grasp failure as spine detachment at any single

contact patch; grasp failure may be catastrophic for a climb-ing robot, motivating this conservative approach. For eachcontact, an established 3D model [16] is used to estimate theloading limit of a linearly-constrained spine array. For bet-ter efficiency we linearize the spine model as a function ofloading direction:

fs = s(v) (13)

where fs is the magnitude of failure force and v is a unit vec-tor defining loading direction. Details of the model are de-scribed in Appendix 9.4 and empirically verified in Section6. For concision, the condition for grasp failure is denotedby f c > f s.

SpinyHand’s grasps typically fail in one of two ways:over-actuating the fingers (internal) or exceeding an allow-able wrench applied at the wrist (external). In the formercase we assume the configuration of the fingers is constant,in the latter case they are subject to compliant motions re-sulting from the external wrench.

5.5.1 Fixed-wrist grasp failureInternal grasp failure is simulated with a fixed wrist po-

sition. Increasing finger tendon forces will eventually causesome of the prismatic phalanges to exceed shear attachment

limits and start to slide across the contact surface. We referto this failure mode as fixed wrist grasp failure. This fail-ure mode also applies if the actuators are controlled to keepthe overall configuration of the fingers constant despite vari-ations in loading.

Because tendon loading angle of the finger tile remainsrelatively constant, the tile preloading force is approximatelyproportional to the tendon force. Simplifying f p(c, ft) asftcp where cp is a scalar, Eqn. 6 can be rewritten:2

f c = A−1[

Krq+ prKpd + pp

]−A−1 ft

[r

cp

]= csys + ftcact

(14)

Finger contact forces f c are a superposition of two vectorcomponents. The system component csys is a constant partof the contact forces caused by passive spring elements inthe design. The actuation component csys is the part in f cthat can be scaled up proportionally by tendon force. As dis-cussed in Section 3, joint spring stiffnesses (functions of Krand pr) are relatively small in the actual design, which causesvectors in csys to have similar directions as cact . The direc-tions of contact forces f c are dominated by cact and remainalmost constant with variations in the tendon force ft , espe-cially when ft is large. By assuming fixed loading angles onthe phalanx spine tiles, the phalanx closest to spine contactfailure (imth phalanx) can be found with Eqn. 13. The max-imum tendon force ft(max) that causes the fixed-wrist graspfailure is solved by iterating Eqn. 14 and 15:

ft(max) =1

‖Rim f (im)c ‖

s(Rim f (im)

c

‖Rim f (im)c ‖

) ft(max) (15)

where Rim is the rotation matrix which transforms contactforce on phalanx im to the local phalanx coordinate frame.The solution typically converges within a few iterations dueto approximate collinearity between csys and cact .

5.5.2 Floating-wrist grasp failureExternal grasp failure is modeled by loading the wrist

with increasing wrenches while keeping finger motors sta-tionary. The wrench moves the wrist slightly, causing smallchanges in the finger poses and varying the contact forces un-til f c > f s at one of the contacts. The corresponding limitingwrench is found by sweeping the wrench direction and mon-itoring the resultant contact forces using the inverse solution(Section 5.4.2) until contact failure occurs (Eqn. 13).

5.6 Grasp Wrench SpaceThe grasp wrench space describes all possible forces

and torques a grasp can generate. When computing sucha space, optimization-based models, such as [20,21,36,38],require extensive computation time to construct the wrench

2 f p is a column vector of 2 if joint and tile pulleys have the same radii

Wang JMR-18-1464 7

Fig. 9: Floating-wrist wrench space computation procedure. (1) Compute (W f i, X f i) for every finger i using the forward approach in Section 5.4. Pointclouds of feasible wrenches for two opposed fingers are constrained by spine tile failure at any phalanx or excessive compressive normal force, Fz relativeto the palm frame. (2) Feasible grasp wrenches are searched with incremental wrist motions, and constrained by hand kinematics. Determine the minimalsize of search space for xg by testing xg sparsely with increasing range until the corresponding x f i of each finger fully covers the solution space X f i. Searchfor every valid wg, such that its corresponding (x f 1, x f 2, . . .x f n) all stay within (X f 1, X f 2, . . .X f n). Sum up every (w f 1, w f 2, . . .w f n) corresponding to thevalid set of (x f 1, x f 2, . . .x f n) to obtain the points cloud that approximates the wrench space Wg. (3) Find the outer boundary of the point cloud using convexhull algorithm so that it can be used to solve maximum wrench limit along any given wrench direction.

space by sweeping over all possible wrench directions. ForSpinyHand, new approaches are possible for the two types ofgrasp failure.

For applications where fixed-wrist behavior is desirable,the tendon force of each finger can be controlled to generatea pre-planned wrench for a given robot motion such that thepose of the fingers with respect to the palm remains fixed.The wrench space in such a scenario is defined by Eqn. 9, 10and 14:

W = {wg|wg = o+n

∑i=1

ftivi, 0≤ fti ≤ fti(max)}

where o =n

∑i=1

Wf icsysi, vi =Wf icacti

(16)

The wrench space of an n-finger hand is a polyhedronspanned by vector vi for each finger, with one of the ver-tex locations at o. For the example of a planar, opposed-finger gripper, the wrench space reduces to a parallelpiped in(Fx,Fz,My) of the hand. When palm contact may apply posi-tive forces to the grasped surface along the z-axis, the wrenchspace becomes a volume swept from the parallelpiped to itsprojection on plane z = 0. More fingers expand the wrenchspace hull to have larger volume and cover more dimensions.

For robots in force-based arm control, SpinyHand canpassively, robustly anchor onto surfaces with stationary fin-ger motors. In this case, the wrench space is limited byfloating-wrist grasp failure. It can be computed using theinverse solution of Section 5.4.2 while searching the wrenchdirection over a 6D unit sphere. This computationally ex-pensive method (O7 of the inverse solution) motivates a newmethod summarized in Fig. 9.

A single finger wrench space point cloud W f can becomputed by searching for limiting finger positions x f corre-

sponding to limiting contact forces:

W f = {w f |w f =Wf f c, f c ≤ f s} (17)

The resulting f c depends only on x f in this approach. Thesearch space should be large enough such that the boundariesof W f are cases of f c = f s. The valid set (defined whenf c ≤ f s) of x f corresponding to W f is denoted X f . For ann-finger hand, a pair of (W f i, X f i) can be computed for eachfinger individually. However, every set of (x f 1, x f 2, . . .x f n)in (X f 1, X f 2, . . .X f n) must satisfy the geometry constraintsof the hand, expressed in terms of the palm. The sum of(w f 1, w f 2, . . .w f n) corresponding to each valid set forms thediscretized representation of Wg. While this new searchproblem requires O6n computation time, it can be reducedto O6 by transforming the search space to xg space (becauseevery x f i is determined by xg). Note that efficiency of thealgorithm is sensitive to size and sparsity of the discretizedsearch space. The intermediate step of finding X f helps re-duce the size of the search space for xg. Populating (W f , X f )with points from (wg, xg) describes every possible wrench onthe finger bases and the palm. In summary, we compute theintersection set of the xg, subject to kinematic constraints,that correspond to the various limiting x f and their associatedfc, and then using the resulting volume of Xg to compute theassociated Wg.

5.7 Contact BreakageThe above model assumes contacts exist on every pha-

lanx and defines grasp failure as spine slippage at any con-tact. At times, higher wrench limits may be achieved ifthe weakest contact’s loading limit is small. For exam-ple, the proximal phalanx can easily lose contact when thehand is loaded away from a large curvature surface, yet theoverall grasp can still succeed. In some cases, such as the

Wang JMR-18-1464 8

Fig. 10: Spine attachment force limit curve of a phalanx spine tile. Eachpoint represents data of at least 5 measurements. The results are presented inpolar coordinates (top) and Cartesian coordinates (bottom). Regions shadedin gray represent compressive (positive) contact forces, and are not of maininterest in this study. The loading angle is defined as the angle between Fxand the loading force direction, while maximum loading force magnitude isdefined as

√F2

x +F2z .

crimp grasp, allowing contact breakage on proximal pha-langes helps form partial closure, however a common load-ing direction on all fingers is then required. If any of thephalanges loses contact, the system of Eqn. 6 becomes over-determined. One of the finger joints may start to move dueto this imbalance until it reaches a joint limit (Fig. 8 (b)).In this case the two adjacent phalanges can be modeled asone combined link and Eqn. 6 can be solved with a smallersquare matrix A. This new reduced equation system is usedto model the crimp grasp in Fig. 8(a).

6 ExperimentsThree sets of experiments help to verify different levels

of model predictions: spine attachment limit, phalanx con-tact forces, and gripper wrench space. All experiments uti-lize coarse 36-grit sandpaper on gripping surfaces in lieu ofrock for consistency and to reduce maximum forces for easeof testing. The maximum force on rock is typically 2-5 timesgreater [17].

6.1 Spine Attachment LimitThe spine contact model is first verified using a single

tile. The tile is placed in contact with sandpaper attached toa force and torque sensor (ATI Gamma), then contact forceis measured as the tile is loaded with increasing force until itslips. As seen in Fig. 10, the empirical data gathered from70 trials match model predictions of the force limit curvein [17]. Gray regions indicate positive normal forces, whereconventional Coulomb friction applies. A comparison of thedata in polar and Cartesian coordinates also suggests that itis reasonable to approximate the limit curve as a straight linein Cartesian coordinates for each phalanx, to simplify char-acterization and computation.

Fig. 11: Experimental setup for measuring phalanx contact forces of agripper with SpinyHand fingers: (1) interchangeable surface with set cur-vature (300mm diameter in the photo); (2) mounting plate for the fingers;(3) grounded board covered in PTFE film; (4) Loading frame with slots toadjust the spring length; (5) signal processing board for the sensors; (6) in-terchangeable contact surface covered with tactile sensors and sand paper;(7) force gauge to measure the loading force (Mark-10, Series 4). The pha-lanx coordinates P and global coordinates G are shown in the figure.

6.2 Phalanx Contact ForcesA second set of experiments measures how contact

forces change on each phalanx while a hand is loaded in thefloating-wrist mode. For these tests, two of the fingers aremounted in a planar frame and made to contact a curved ob-ject, as seen in Fig. 11. The frame holding the fingers slideson a low-friction PTFE film as the grasp adjusts to changesin loading. Loads are applied by pulling on a cable attachedto the frame, producing forces, Fx,Fz. A spring attached toeach tendon provides an approximately constant grasp forcefor small motions.

Taking advantage of symmetry, the grasp is instru-mented by having each phalangeal tile on the left fin-ger contact an instrumented block with a thin, capacitiveforce/torque sensor adapted from [39]. The sensors measurecontact forces and moments at 109 Hz with an accuracy ofapproximately 5% for 0-25 N in Fx and 0-10 N in Fz.

The testing procedure is: (i) place the fingers aroundthe curved object and align each tile on the correspondingcontact block; (ii) preload the gripper with tendon forces of37 N; (iii) pull on the external loading cable while measuringthe force with a force gage (Mark-10, Series 4, 157 Hz) untilany tile slips.

As shown in Fig. 12, shear and normal forces measuredat the contacts match trends predicted by the model. As thegripper initially grasps the surface, each phalanx tile slidessome distance before catching on asperities. Therefore eachphalanx starts with a different shear preload. As the exter-nal loading force pulls the hand away from the surface, shearcontact forces increase gradually, with some oscillation dueto friction in the pulley systems, as discussed in Section 5.2.Results also show that the normal components at the proxi-mal and distal phalanges decrease, so that these contacts arelikely to exceed their force limits. The middle phalanx typi-cally stays within safe loading conditions. The exact failurecondition depends on the details of spine-asperity engage-ment at each tile and varies somewhat on a trial by trial basis.

The grasp model is linear and does not explain some

Wang JMR-18-1464 9

Fig. 12: Variation of shear (Fx) and normal (Fz) contact forces on each of the phalanges as the gripper is loaded with increasing force along the negativeglobal Fz direction. Model predictions in the left two plots show similar trends as the actual measurements in the right two plots.

Fig. 13: Wrench space experiments and model predictions for grasping on a 300mm diameter curved surface. The wrench space hull in gradient color isnumerically computed with the grasp model. The black dots are average results of 5 measurements along each wrench direction vector. The red spheredisplays the standard deviation of loading force magnitude for each data point. The dots that are not covered with red spheres are mirrored data to show thecomplete wrench space. Prediction error on average is 10%.

observed nonlinearity in the plots, which arises from a com-bination of backlash and friction in the system. In addition,during these experiments, the gripper base shifts by up to5mm, consistent with model predictions. However, in themodel we assume that corresponding changes in joint an-gles, tendon length, and loading spring force are negligible.This is counter to the ill-conditioned property of A in Eqn.6 (Section 5.1), which suggests that small changes in fingerpose (q) can significantly alter contact forces.

6.3 Gripper Wrench Space

The experimental setup and procedure for measuring thewrench space is similar to that in Fig. 11. However, the exter-nal load cable is now attached to various points on the frameto produce a force an moment (Fx, Fz, My). The pulling di-rection is held constant as the force increases until either oneof the tiles slips or the entire hand detaches. For a 300 mmdiameter surface, 17 unit vector directions were selected toevenly cover the wrench space for Fz ≤ 0 (i.e., when pullingthe object away from the fingers). Five trials were repeated

Wang JMR-18-1464 10

for each load location. The sandpaper patches were replacedevery 25 trials to reduce the effect of contact surface degra-dation. A lower curvature surface with a 700 mm diameterwas additionally tested.

As seen in Fig. 13 for the 300 mm diameter curved sur-face, the negative Fz region of the wrench space has an in-verted “mountain” shape. We assume symmetry in the modeland reflect the empirical data points to cover the space. Whenone applies some positive shear force Fx, one side of the fin-ger tiles has larger shear forces and the other side has re-duced shear forces. Accordingly, the side in the +x directionis closer to losing contact whereas the opposite side is not indanger of losing contact and has some margin with respectto its spine adhesion limits. Hence adding a positive momentto keep both sides firmly in contact can actually increase theoverall strength of the grasp with respect to pull-out forces.

The average failure conditions match the model, al-beit with variability in some cases, denoted by the radiusof the red shaded region around each empirical data point.The overall deviation from the model averages 10%, and isgreatest in high-wrench regions where trial-to-trial variabil-ity is highest. Grasping tests on the lower curvature surface(700 mm diameter) were conducted using two different fin-ger tendon actuation forces, 30N and 40N; an average modelerror of 12% and 14% resulted respectively. In either case,the limit surfaces are smaller (it is harder to grasp a largediameter object) and the errors are somewhat larger.

7 ResultsThe model can predict where phalanx contact failure

will likely occur first. Hence it could be useful for graspand load planning. Figure 14 visualizes the failure modeswith different color dots covering the wrench space surfaces.Grasping on more curved (smaller radius) surfaces results ina larger wrench space, and most contact failures start fromthe proximal phalanx. Distal phalanx failure occurs whenthe loading moment, My, is comparatively small comparedwith shear loading force, Fx. When grasping a large radiussurface the entire wrench space is dominated by proximalphalanx failure. These results match experimental observa-tions, and indicate that initial grasp slipping may be reducedby enhancing the proximal phalanx attachment, e.g. with adenser array of spines.

SpinyHand has a skew symmetric wrench space, for thisprototype (when loaded in the plane of opposed finger ac-tuation). One can optimize grasp attachment strength, in-terpreted here as either

√F2

x +F2z or My, by changing the

applied wrench direction. Figure 14 shows that for both sur-face curvatures, a higher shear force to moment ratio Fx:Myimproves point force attachment (optimal ~90:1, dashed),while a lower ratio achieves better moment resistance (op-timal ~20:1, dotted). The dashed and dotted lines representloading planes (wrench slices) that respectively correspondto the top and side views in Fig. 13. These insights caninform robot motion planning to distribute loads effectivelyamong different end-effectors and reduce the risk of fallingwhile climbing. In addition to active control strategies, the

Fig. 14: Visualization of failure modes and loading configuration on twograsping conditions (300mm and 700mm diameter surfaces) that have beenverified with experiments. Red and blue dots represent proximal and distalphalanx failure (PP and DP) respectively. The dash lines and dotted linesindicate optimal loading planes to achieve large resisting force and momentrespectively.

Fig. 15: Shear and normal grasp force limits for the two previously dis-cussed surfaces with changing finger tendon forces. Blue lines: 300 diam-eter surface. Red lines: 700 diameter surface. The shear force is underloading configuration of Fx = 50My.

hand mechanism can be tailored to task requirements. Forexample, if the hand is loaded at a single point via a cable(as in these studies), the best place to anchor the loading ca-ble for the 300 mm diameter surface is 1.1 cm away from thepalm surface along the center axis of the palm (negative zdirection) to produce an ideal combination of force and mo-ment, based on the results in Fig. 14.

7.1 Parameterized StudiesMany factors influence grip strength, including surface

properties, finger kinematics, and actuation control strategy.We can use the previous grasp model in simulations with var-

Wang JMR-18-1464 11

Fig. 16: Shear and normal grasp attachment force limits for the two pre-viously discussed surfaces with changing middle and distal phalanx pulleyradii and constant finger tendon force (46N). The color map shows the forcemagnitudes. Surface A and B are 300mm and 700mm diameter surfaces.The blue region indicates that phalanx contact failure has occurred beforeapplying external load. The dark blue blocks are cases where the gripperejects itself away from the surface. The white diamond is the pulley radiiselections of the implemented SpinyHand.

ious hand configurations to investigate how finger actuationand transmission ratios influence grasping.

Grasp experiments maintain a constant finger tendontension of 37 N. To investigate the effect of different ten-don tensions, a batch of wrench simulations were run forthe opposed two-finger gripper on the curved 300mm and700mm diameter curvatures, now covered in fine concretesurfaces (higher spine engagement strength [17]). A rangeof finger tendon forces were tested under the loading config-uration Fx = 50My. As shown in Fig. 15, moderate fingertendon forces help expand the wrench space. Large tendonactuation may help obtain better normal attachment on lowercurvature surfaces, but gradually reduces shear loading ca-pability. In general, the wrench space for a low curvaturesurface is both lower and less sensitive to tendon force varia-tions. On more curved surfaces, the normal force componentof the finger contact contributes substantially to attachment.However, it also can lead to premature failure at the distal orproximal phalange. In general, controlling the tendon forceis more important for objects with higher curvature.

The SpinyHand fingers contain numerous load-sharingelements that can be adjusted, including joint stiffnesses andpreloads, prismatic phalanx stiffnesses and preloads, andjoint and tile pulley radii. Based on Eqn. 14, the system com-ponent csys determined by the spring elements contributes toonly the initial states of the grasp. As finger tendon forceincreases, the actuation component cact will eventually dom-inate the contact forces. For a given tendon tension, this cactdepends only on the pulley radii r of each finger joint andthe tile pulley radii which determine f p. These are arraysconsisting of n scalars for an n-phalanx finger. In the fingerplane, tendon force directly determines shear components of

the contact forces; the corresponding normal force compo-nents are affected by varying these design parameters. Con-sidering the inverted matrix A, change in only one of the ndegree of freedom variables r and f p will adjust every con-tact force. As it is practically easier to adjust the pulley radiivector r (as compared to f p), they are considered here as adesign parameter for optimizing grasp performance. The re-maining design parameters remain useful for tuning fingerbehaviors, such as the closing sequence.

Wrench simulations were conducted while varying mid-dle and distal joint pulley radii. The proximal pulley radiusis fixed to be 5.5 mm, as in the SpinyHand prototype. Theradius of the distal joint pulley should not be larger thanthe middle joint pulley, as this condition could lead to fixed-wrist grasp failure due to finger actuation (dark blue regions).Lowering the tendon force may help decrease the size of thefailure regions, but also reduces the attachment force limitaccording to Fig. 15. In order to achieve adequate perfor-mance on both the 300 mm and 700 mm surfaces, SpinyHandemploys pulley radii represented by white diamonds in Fig.16.

These results provide initial insights into how design andcontrol can be influenced by an understanding of multifingerspine attachment for rock climbing.

8 ConclusionThis work introduces a multi-finger hand designed for

human-scale robots climbing on steep, rocky terrain. In-spired by human rock climbing techniques, the hand can per-form a variety of grasp types, such as sloper or pinch grasps,or cling to an edge with its fingernails. Dense arrays of spinesimprove contact properties at each phalange and in the palm.The tiles are connected by tendons and float on linear tracks,ensuring load-sharing along each finger. The phalange de-sign particularly improves gripping robustness on challeng-ing low-curvature surfaces by allowing each contact to sup-port large shear loads. A modest amount of transmission fric-tion can actually be helpful because it allows those tiles withthe strongest and stiffest contacts to take slightly larger loads.

Contact force and wrench space grasp models are devel-oped, taking advantage of the load-sharing among phalangesto greatly accelerate the solution. A fast wrench-space com-putation method is presented, searching for the space of lim-iting compliant motions of the wrist, subject to kinematicconstraints, and then computing the corresponding forces.Although it is challenging to test the entire hand to fail-ure due to the very high forces involved, experiments wereconducted with two fingers in a symmetric planar grasp us-ing measured applied loads and contact forces. The resultsmatch numerical predictions, with some variability due tofriction, backlash and a modeling assumption that small mo-tions do not significantly affect the kinematics of the fingers.

Simulations of the hand on small- and large-curvatureobjects are presented, and allow one to examine differenttendon actuation strategies and the effects of varying designparameters such as pulley radii. In general, the wrench spaceof allowable external loads becomes smaller as surfaces be-

Wang JMR-18-1464 12

come increasingly flat and the sensitivity to variations in ac-tuating tendon tension become less.

8.1 Future workWhile the grasp model is generalized for 6D wrenches,

experimental validation has examined only planar grasps.Future work will investigate intermediate orientations of therotated outer fingers (instead of just 0o or 180o). As the graspmodel continues to be validated in these more general grasps,it also becomes useful to incorporate the grasp model intoa package such as SimGrasp [40] exploring the effects ofchanging phalange lengths, stiffnesses, etc.

Adjustments to the hand design could improve capabil-ity and resilience. For example, there may be phases dur-ing grasping when the spines should be retracted to induceslipping contact temporarily. Pneumatic retraction of thespines could improve conformability and spine/gripper life-time by decreasing unnecessary spine engagement. A com-pact clutch or brake design could additionally reduce thenumber of motors and overall weight of the hand [41], whileincluding tendon tension sensors and fingerpad tactile sen-sors could enable new closed-loop control strategies. Theseupdates could enable innovative gripping techniques specificto rock climbing.

The ultimate goal is to integrate this technology with theRoboSimian quadruped, resulting in new questions regardingintegration of design, control, and sensitivity.

AcknowledgmentThis work is supported by the National Science Foun-

dation (Award IIS-1525889). Any opinions, findings, con-clusions or recommendations expressed in this material arethose of the authors and do not necessarily reflect the viewsof the funding sources.

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[41] Aukes, D. M., Heyneman, B., Ulmen, J., Stuart, H.,Cutkosky, M. R., Kim, S., Garcia, P., and Edsinger,A., 2014. “Design and testing of a selectively com-pliant underactuated hand”. The International Journalof Robotics Research, 33(5), pp. 721–735.

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9 Appendices9.1 Design Details

As shown in Fig. 17, four brushed motors (Maxon RE2512V) are housed inside the two halves of the palm cham-ber. Double bronze bushings support each motor shaft thatwinches a tendon to actuate each finger. Two small, brushedmotors (Pololu, 0.22Nm, 320RPM, 12V) with worm drives(1:48) alter the alignment of the two outer fingers. The ro-tary finger base assembly (Fig. 18) also includes an angularcontact bearing to resist finger moments and a thrust bearingto reduce friction caused by axial mounting force. A chan-nel through the assembly is left for the finger tendon to comeout of palm chamber. This through hole is concentric withthe rotary motion axis to prevent tendon tangling.

The hand is equipped with eight analog sensors to mon-itor the four tendon pulley positions as well as positionsand moments on the two rotary finger bases. Tendon pul-ley positions are tracked with four magnetic rotary encoders(RLS RMB20) installed on the back shaft of the big motors.Two pairs of magnets (hollow cylindrical for tendon to gothrough) and hall effect transducers are directly mounted onthe worm gear to track absolute position of the rotary fin-ger base. To prevent large impacts and torques transmittedfrom the finger from damaging the worm gear, a cubic mag-net is installed on each of the small motor mounting plates(Fig. 17(g)) to sense the plate deflection, and thus measurethe axial force created by the worm gear transmission. Themotor will backdrive when torque is above certain thresholdto prevent damage. The sensors are signal-conditioned by acustomized four-layer PCB board (Fig. 17(c)) and read by amicro-controller (Teensy3.2) stacked on the board.

Besides sensor signal processing, the PCB board inter-faces with four external motor drivers (15A nominal, for eachbig motor) and a dual channel motor driver (0.7A nominal,for each small motor). Power for logic and motors is regu-lated and transmitted through a separated region on the mid-dle layers of the board. The compact electronics package fitsinside the bottom palm chamber.

9.2 Palm Model DerivationsForce balancing equations on the tiles (Fig. 5) indicate

tendon travel tendency, thus friction direction, is unknown:

{Ft0 +Ft1 + kp(x− x1) = k1x1

Ft1 +Ft2 + kp(x− x2) = k2x2(18)

If the tendon is static and we assume contact stiffness k1 > k2,x2 will be larger than x1 and the tendon has the tendencyto travel to the left. The force balancing equations on the

Fig. 17: Actuation and electronics design in the palm chamber: (a) doublebushings shaft support for bottom chamber motors; (b) double bushing shaftsupport for top chamber motors; (c) 4-layer PCB board; (d) small motorand worm drive assembly; (e) big motors; (f) mounting plates for hall effecttransducers to measure rotary finger position and torque; (g) cubic magnetfor torque sensing and (h) worm gear mounted to the finger base and magnetfor its position sensing. More details about the 4-layer PCB board are infront view (c1): (i) sensor signal conditioning circuits, (ii) Teensy 3.2 micro-controller, (iii) power regulator, (iV) dual channel small motor driver and inrear view (c2) where (v) are four stacked big motor drivers.

Fig. 18: Rotary finger base design: (1) worm gear; (2) thrust bearing; (3)compact cross-roller bearing (transparent mode); (4) finger base adaptingplate; (5) finger base phalanx. The rectangular areas with dotted line showthe solid part of the palm wall pinched by the finger base assembly.

pulleys becomes:

{(1+µ/2)Ft1 = (1−µ/2)Ft0

(1+µ/2)Ft2 = (1−µ/2)Ft1(19)

Both ends of the tendon are fixed to the palm, and dis-placements of the palm and two tiles should satisfy: x1+x2 =2x. Total force on the palm is the sum of contact forces oneach tile F1 = k1x1 and F2 = k2x2, and can be solved usingthe above equations.

9.3 Tendon FrictionWhen the hand is settled statically on a surface, the ten-

don cannot freely slide around the joint pulleys due to fric-tion; the local tension varies along the finger tendon untilforce exceeds static friction. We assume the tendon tensionis constant along the portion of tendon within the same pha-lanx. As a loading force on the wrist is applied, the hand

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configuration xg and finger joint positions q will change. Ifthe spine tile remains stationary with respect to the phalanx,it has to travel with respect to the surface. The displace-ment of tile tangential to the surface on phalanx i is denotedby δxci. However, as finger joints rotate slightly, the jointpulleys either wind or unwind the finger tendon. The re-sulted variation on local tendon length also changes the spinetile position with respect to the phalanx by δlci. The actualtile displacement with respect to the contact surface becomes(δxci + δlci). Tile displacement deforms the spines and sur-face in shear, inducing an additional change in tangentialcontact forces on phalanx i:

δ fci = (δxci +δlci)kci (20)

where kci is the contact stiffness between the phalanx frame.Surface compliance is empirically characterized.

9.4 Simplified Spine Model3D spine tile attachment force limits are captured with

probability-based model [16] solved with the Monte CarloMethod. For better efficiency, we simplify the contact modelbefore integrating it to the grasp model. As empiricallyfound in [16,17], attachment is approximately linear with theloading angle. This property allows the limit curve on thesame loading plane to be linear fitted:

{fs = fsmax− ksl pφ;ksl p = ( fsmax− fsmin)/φmax;

(21)

where fsmax and fsmin are the maximum and minimum empir-ical tile attachment, φ is the force loading angle on the spinetile and φmax is the loading angle corresponding to fsmin.

10 Listing of figure captionsFigure 1 on page 2: SpinyHand grasping a pumice rock.The contact areas are covered in microspines to support largetangential forces.Figure 2 on page 3: (left) SpinyHand CAD rendering shows22 spine-laden tiles in yellow distributed among 4 indepen-dent fingers and a palm. (right) Rock climbing techniques(a-d): (a) sloper grasp on a gently curved surface; (b) openhand/half crimp grasp on an edge; (c) pinch grasp on a nar-row feature; (d) pose between a sloper grasp and pinch. (d) isnot commonly used by human rock climber but can be help-ful for SpinyHand to envelop large-curvatures.Figure 3 on page 3: Spine tile design: (1) sliding block forprismatic motion of the finger spine tile along x axis; (2)fixed pulley (horizontal metal rod); (3) groove for tendon

routing; (4) sliding channel for palm tile; (5) fixed pulley(vertical metal rod). The red arrow indicates the sliding andloading direction. The tile coordinates are consistent withthe finger phalanx coordinates P.Figure 4 on page 4: Diagram and pictures of the finger: (1)fingernail; (2) spine tile; (3) joint pulley; (4) phalanx exten-sion spring; (5) phalanx frame; (6) phalanx pulley; (7) jointextension spring. Ft is the tendon force. The tendon (red)routes around each joint pulley and phalanx pulley, so that Ftsimultaneously curls the finger and slides the spine tile alongthe phalanx.Figure 5 on page 4: Left: Palm experiment setup. Right: di-agram of a 2-tile system. Parts with boundaries or shadesin the same color are mounted together as single movingparts. Two tiles in gray are preloaded by springs (stiffness kp)mounted to a moving palm frame in red. The tiles have con-tact stiffness k1 and k2 with respect to the ground in blue. Atendon routes over pulleys on the tile (gray) and palm frame(red). Local tendon tensions Ft1 and Ft2 may be different dueto friction on pulleys between adjacent tiles. x1, x2 and xare global displacements of the tiles and palm frame respec-tively.Figure 6 on page 5: Predicted and experimental maximumload of a two-tile system. Horizontal axis shows the contactspring configurations (springs 1, 2 and 3 have stiffness of 1.4,2.2 and 4.9N/mm respectively). Five measurements are con-ducted for each spring configuration. The percentage in redshows how much friction can improve the loading capacity.Figure 7 on page 5: Empirical maximum load of a four-tilesystem with and without friction. Horizontal axis shows thecontact stiffness spring configurations, with the spring typesequence starting from the left.Figure 8 on page 5: Three coordinates are used in the handmodel: 1) palm (global) coordinates G; 2) finger coordinatesF ; 3) phalanx coordinates P. Blue dots indicates the contactlocations. ci and qi are contact forces and joint angles. Athird finger is behind the grasping surface and therefore dis-played with dotted lines. The top diagrams show: (a) crimpgrasp with only distal phalanx contact; (b) pinch/sloper graspwith proximal contacts broken.Figure 9 on page 8: Floating-wrist wrench space compu-tation procedure. (1) Compute (W f i, X f i) for every fingeri using the forward approach in Section 5.4. Point cloudsof feasible wrenches for two opposed fingers are constrainedby spine tile failure at any phalanx or excessive compres-sive normal force, Fz relative to the palm frame. (2) Feasiblegrasp wrenches are searched with incremental wrist motions,and constrained by hand kinematics. Determine the mini-mal size of search space for xg by testing xg sparsely withincreasing range until the corresponding x f i of each fingerfully covers the solution space X f i. Search for every valid wg,such that its corresponding (x f 1, x f 2, . . .x f n) all stay within(X f 1, X f 2, . . .X f n). Sum up every (w f 1, w f 2, . . .w f n) corre-

Wang JMR-18-1464 16

sponding to the valid set of (x f 1, x f 2, . . .x f n) to obtain thepoints cloud that approximates the wrench space Wg. (3)Find the outer boundary of the point cloud using convex hullalgorithm so that it can be used to solve maximum wrenchlimit along any given wrench direction.Figure 10 on page 9: Spine attachment force limit curve ofa phalanx spine tile. Each point represents data of at least 5measurements. The results are presented in polar coordinates(top) and Cartesian coordinates (bottom). Regions shadedin gray represent compressive (positive) contact forces, andare not of main interest in this study. The loading angle isdefined as the angle between Fx and the loading force direc-tion, while maximum loading force magnitude is defined as√

F2x +F2

z .Figure 11 on page 9: Experimental setup for measuring pha-lanx contact forces of a gripper with SpinyHand fingers: (1)interchangeable surface with set curvature (300mm diameterin the photo); (2) mounting plate for the fingers; (3) groundedboard covered in PTFE film; (4) Loading frame with slotsto adjust the spring length; (5) signal processing board forthe sensors; (6) interchangeable contact surface covered withtactile sensors and sand paper; (7) force gauge to measure theloading force (Mark-10, Series 4). The phalanx coordinatesP and global coordinates G are shown in the figure.Figure 12 on page 10: Variation of shear (Fx) and normal(Fz) contact forces on each of the phalanges as the gripper isloaded with increasing force along the negative global Fz di-rection. Model predictions in the left two plots show similartrends as the actual measurements in the right two plots.Figure 13 on page 10: Wrench space experiments andmodel predictions for grasping on a 300mm diameter curvedsurface. The wrench space hull in gradient color is numer-ically computed with the grasp model. The black dots areaverage results of 5 measurements along each wrench direc-tion vector. The red sphere displays the standard deviationof loading force magnitude for each data point. The dots thatare not covered with red spheres are mirrored data to showthe complete wrench space. Prediction error on average is10%.Figure 14 on page 11: Visualization of failure modes andloading configuration on two grasping conditions (300mmand 700mm diameter surfaces) that have been verified withexperiments. Red and blue dots represent proximal and distal

phalanx failure (PP and DP) respectively. The dash lines anddotted lines indicate optimal loading planes to achieve largeresisting force and moment respectively.Figure 15 on page 11: Shear and normal grasp force lim-its for the two previously discussed surfaces with changingfinger tendon forces. Blue lines: 300 diameter surface. Redlines: 700 diameter surface. The shear force is under loadingconfiguration of Fx = 50My.Figure 16 on page 12: Shear and normal grasp attachmentforce limits for the two previously discussed surfaces withchanging middle and distal phalanx pulley radii and constantfinger tendon force (46N). The color map shows the forcemagnitudes. Surface A and B are 300mm and 700mm diam-eter surfaces. The blue region indicates that phalanx contactfailure has occurred before applying external load. The darkblue blocks are cases where the gripper ejects itself awayfrom the surface. The white diamond is the pulley radii se-lections of the implemented SpinyHand.Figure 17 on page 15: Actuation and electronics design inthe palm chamber: (a) double bushings shaft support for bot-tom chamber motors; (b) double bushing shaft support fortop chamber motors; (c) 4-layer PCB board; (d) small mo-tor and worm drive assembly; (e) big motors; (f) mountingplates for hall effect transducers to measure rotary finger po-sition and torque; (g) cubic magnet for torque sensing and(h) worm gear mounted to the finger base and magnet for itsposition sensing. More details about the 4-layer PCB boardare in front view (c1): (i) sensor signal conditioning circuits,(ii) Teensy 3.2 micro-controller, (iii) power regulator, (iV)dual channel small motor driver and in rear view (c2) where(v) are four stacked big motor drivers.Figure 18 on page 15: Rotary finger base design: (1) wormgear; (2) thrust bearing; (3) compact cross-roller bearing(transparent mode); (4) finger base adapting plate; (5) fingerbase phalanx. The rectangular areas with dotted line showthe solid part of the palm wall pinched by the finger baseassembly.

11 Listing of table captionsNo tables are included in this paper.

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